The TV presenters also said three White House officials had told them the National Enquirer would publish a negative article about them unless they apologised to Mr Trump for their scrutiny of him.

Mr Scarborough said: “They said, ‘if you call the president up and you apologise for your coverage then he’ll pick up the phone and basically spike the story.'”

“I had, I will just say, three people at the very top of the administration calling me.”

Ms Brzezinski said reporters from the supermarket tabloid began harassing her family.

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“They were calling my children,” she said. “They were calling close friends.

“These calls persisted for quite some time and then Joe had the conversations that he had with the White House where they said ‘oh, this could go away’.”

David Pecker, chief executive of the Enquirer’s parent company, is a staunch ally of Mr Trump, which is reflected in the tabloid’s coverage of the president.

But in a Friday morning tweet, Mr Trump denied the couple’s claims, saying Mr Scarborough “called me to stop a National Enquirer article. I said no!”

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Media captionThe MSNBC hosts say the White House demanded an apology for coverage, or the National Enquirer would run a story on them

Mr Scarborough swiftly hit back: “Yet another lie. I have texts from your top aides and phone records. Also, those records show I haven’t spoken with you in many months.”

Mr Scarborough also said that a “well-known congressman” had told him that the president went on a “rant” against the show during a meeting with around 20 legislators at the White House to discuss efforts to pass healthcare legislation.

“I’ve been in politics my whole life. … He scared me,” Mr Scarborough said, describing what the source had told him.

“Because he was vicious when he turned from you to Mika,” he described being told.

“His face was red. He started talking about blood coming out of her ears, out of her eyes.”

The Enquirer’s owner said on Friday morning the tabloid had published a story at the beginning of June about the couple’s relationship “the truth of which is not in dispute”.

Dylan Howard, vice-president of the publication’s parent company, added in the statement: “We have no knowledge of any discussions between the White House and Joe and Mika about our story, and absolutely no involvement in those discussions.”

The Enquirer’s Twitter account retweeted their article on Friday, with a headline about “Morning Joe Couple’s Sleazy Cheating Scandal!”

Mr Trump’s original tweet about the MSNBC hosts provoked a storm of criticism, including even from some fellow Republicans.

Senator Lindsey Graham said Mr Trump’s remarks on Thursday were “beneath the office” of president.